stevendaryl said:
Perhaps I should start a separate topic for White Holes, but I really don't understand why there is a white hole region.
How do we tell the past from the future?
The textbook treatments are all so dry that I skipped over them rather lightly. I suppose you'd want to read up on "time orientable manifolds" if you wanted the formal description of how to do this. (Wald would have this).
Informally, let's start with assuming one knows how to construct light cones. Note that one has to be careful about this inside the event horizon if one is using Schwarzschild coordiantes!
It's easy enough to determine the two light-like geodesics that pass through a point, and draw the cone shape that light marks out. But if one draws a point P, one needs to realize that the Lorentz interval between P and P+dt is spacelike inside the event horizon. Which implies that the correct "shading" of the light cone to determine its "inside" region does not include the point P+dt inside the event horizon - given the convention that we "shade" the light cone so that the inside (shaded) region only contains timelike worldlines.
Basically, we know that P+dr and P-dr are both timelike intervals inside the event horizons, so both of those are in the "shaded" region, and P+dt and P-dt are not in the shaded region.
So, onece we've got the easy part done, shading the light cone correctly so that it only contains timelike worldlines, we still need to determine past vs future.
As far as I know, the only way to do this is by convention, given that physics is time reversible. So you pick some external observer, and say that as the Schwarzschild t increases at large R, that that is the future.
Then you need to splice all the light cones together in a consistent manner. This is the tricky part. There's really only two choices inside the horizon in Schwarzschild coordinates though - r increasing and r decreasing. It turns out that in the black hole region it's r decreasing, in the white holde region it's r increasing.
Its probably easy to demonstrate this by using KS coordinates, where the light cones always point in the same direction , than it is to demonstrate in Schwarzschild coordinates (where they rotate). You'll probably need some non-singluar coordinate system to convicingly handle the transition over the horizon in any event.